If there are two interacting subsystems A and B, how to define the information flow from A to B in quantum regime and classical regime?
One way to quantify the flow of quantum information under evolution by a unitary operator is the so-called flow or index of the unitary, as in
Kitaev, Anyons in an exactly solved model and beyond (2006), https://arxiv.org/pdf/cond-mat/0506438.pdf
Gross, Nesme, Vogts, Werner, Index theory of one dimensional quantum walks and cellular automata (2012), https://arxiv.org/pdf/0910.3675.pdf
In classical physics information flows between systems when the measurable quantities of one system depend on those of another, e.g. the frequency at which a weight bobs on the end of a spring will depend on the Young's modulus of the spring.
In quantum physics, information flow can be characterized by dependence of the observables of one system on those of another. The observables of one system can depend on those of another although it is impossible to tell by measuring either system alone: this is called locally inaccessible information.